To Numbers and Back Again

This puzzle is about the classical Indian language Sanskrit. The puzzle consists of a meta followed by four minis. As clued by the global flavortext, each mini will be associated with an old scheme for converting symbols to numbers and ultimately yield six numbers. In the meta we will use these 24 numbers to recover Sanskrit sounds.

Shifting and Stepping

The puzzle is set in a complex of rooms connected in two ways: via archway and via staircase. Reading along the first letters of each sentence gives the cluephrase “GRAHA BHEDAM HALF STEP”. The first part of the cluephrase, Graha bhedam, is an operation in Carnatic (South Indian classical) music which takes a raagam (analogous to a mode in Western classical music) to another raagam by shifting the tonic. Given the overall Indian theming of the puzzle, this clues that the puzzle is based on the Melakarta raagam system of Carnatic (South Indian Classical) music. Each room in this complex corresponds to one of the 72 Melakarta raagams. The two ways of passing from room to room: via archway and via staircase - correspond to the two parts of the cluephrase, graha bhedam from the current raagam (shifting) and going to a raagam which differs from the current one by a half step (stepping). Each melakarta raagam has seven notes, so going up seven levels in the staircase takes you back to the current room. This is confirmed by the drone, which represents the tonic - it changes when going up and down levels of the staircase, but not when going between rooms via archway. Several rooms have environmental descriptors. These tell you what chakram the raagam is in:

• The hot rooms are in the Agni (fire) chakram;
• The room with a storm is in the Rudra (god of wind/storms) chakram;
• The room with arrows is in the Bana (bow) chakram;
• The moonlit room is in the Indu (moon) chakram;
• The rooms with art of the Big Dipper are in the Rishi chakram, as the Big Dipper in Sanskrit is the “seven sages”;
• The rooms with a 10 pointed “compass rose” are in the Disi (direction) chakram.

We can now go line by line. Throughout, we use the Indian solfege notation S R G M P D N S (analogous to do re mi fa so la ti do). We refer to raagams by their index in the list of Melakarta raagams - this index comes from the Katapayadi system, as the name of the raagam encodes its index in the 72 (which are ordered lexicographically). Two raagams are said to be neighbors if they differ by a half step in a single scale degree.

Sorting in rainbow order, the answer to this puzzle are the numbers 13, 19, 19, 26, 37, 39.

Val’s Storyboard

The title of the puzzle, “Val’s Storyboard”, is a reference to Valmiki, the author of the Ramayana. Parts of the Ramayana can be referred to by kanda (book), sarga (chapter), and sloka (verse). Each image depicts a particular sloka of the Ramayana. Through reading the story, or possibly brute force, we can identify the slokas corresponding to the scenes. The bracketed numbers then specify which characters of these slokas to take, and by doing so each image yields a word. Each word corresponds to a number in the Bhutasamkhya system–which generally associates the name of a set with the size of the set. We refer to the examples on Wikipedia as well as this list to obtain a number for each image.

Plugging these into the given mathematical equations, we end up with the numbers 1, 2, 3, 11, 16, 24.

Surfing Through Time and Space

We are provided with a sorted list of unix time timestamps. Taking last two digits of the timestamp and applying A1Z26, spell out the cluephrase “PARTITION DOTW THEN DRAW SEGMENTS BETWEEN NAKSHATRAS ON C SPHERE”. Every timestamp is on one of the five weekdays, and after partitioning by day of the week, there are exactly two timestamps per year. Each timestamp can be associated with a nakshatra - a lunar mansion. Roughly, the 27 nakshatras divide the orbit of the moon - which is 360 degrees - into 27 sectors, and each nakshatra lies in one of the sectors. Thus, each time can be associated with a nakshatra, and there are numerous online calculators (nakshatras are still a major part of modern Hindu astrology, and come up in most Hindu religious ceremonies) for doing so. Note that the nakshatra - which corresponds to the position of the moon relative to the stars - depends only on the absolute time - these sites ask for location only to adjust the input time for time zone. In any case, after partitioning by day of the week and finding nakshatras, we obtain the following.

We can now draw lines on the celestial sphere - the two nakshatras in each year give us a line segment. In what follows, the left and right disks are the Northern and Southern celestial hemispheres.

Monday

Tuesday

Wednesday

Thursday

Friday

Each of these constellations is a (Western) zodiac symbol. The zodiac constellations (probably) predate both Hellenic and Vedic astronomy, and both use the same zodiac symbols. Thus, we obtain the following.

We now look at the six lines at the bottom of the puzzle. There are five distinct Devanagari characters which appear: सो, म, बु, बृ, शु. These are the first letters of the Sanskrit words for Monday, Tuesday, Wednesday, Thursday, and Friday, respectively.

We can apply the Aryabhata numeration scheme - which was traditionally used in astronomical texts - to associate the characters in the above constellation names with numbers.

बृ[1] (mod 123) = 19 (mod 123) = 19

म[3] म[2] शु[1] (mod 123) = 20 + 170000 + 60000000 (mod 123) = 19

सो[1] (mod 123) = 25 (mod 123) = 25

बु[2] बृ[2] शु[1] (mod 123) = 50 + 200000 + 60000000 (mod 123) = 37

शु[3] म[2] (mod 123) = 24 + 170000 (mod 123) = 38

सो[1] म[1] बु[1] (mod 123) = 25 + 2500 + 160000 (mod 123) = 42

Thus, the answer to this minipuzzle are the numbers 19, 19, 25, 37, 38, 42.

Traditional Chess Puzzles

As clued by the flavortext, these are seriesmover puzzles, where White must make two moves in a row with no reply from Black, though the first move cannot be a check. At the end of this series of moves, Black should be checkmated or otherwise defeated.

However, these puzzles are not generally solvable under regular chess rules. In addition, the boards look strange, with no checkered pattern and with X’s on some squares. Given the title of this minipuzzle and the presence of Devanagari letters, we might deduce that we are dealing with the ancient Indian variant of chess called Chaturanga.

In Chaturanga, the king, rook, and knight have the same moves as in modern chess. The pawn mostly moves the same too, though it doesn’t have an initial double-step move. There are a few possible variants of promotion rules, but here we’ll use the version where the pawn must promote to the piece originally occupying the promotion square. The queen moves one step diagonally, while the bishop moves two spaces diagonally, while being able to jump over any piece in the square in between (this is the first of the 3 potential movement rules described on Chaturanga’s Wikipedia page, and the one that is most commonly seen in other sources). We will also assume the normal check and checkmate rules, though a player can also win by baring the opponent’s king, i.e., by capturing all enemy pieces except the king.

For each of the 6 Chaturanga puzzles, we find the squares White moves into for their first and second moves. Finding the corresponding Devanagari letters or syllables on the extraction board at the end, we convert them to numbers according to the Aksharapalli scheme (which is hinted at by the page numbers). We then take the products of each pair of numbers to extract 6 numbers in ascending order: 1, 6, 16, 25, 35, 42.

Meta

The colored rectangles in each mini title overlap with the phrase “Shiva Sutra”, which refers to Panini’s 14 aphorisms which give an ordering of Sanskrit sounds (this is disambiguated from the other Shiva Sutras by the flavor text). Note that the Shiva Sutras do not include every possible Sanskrit phoneme, only the building blocks for all phonemes. Once solvers have the numbers from all the minis, they index into the Shiva Sutras to obtain the following:

Each box is a single Sanskrit character, so sounds are collapsed in the “natural” way to form characters. There are two diacritics which appear. The first is an anusvara, which adds an “-m” to the end of a syllable. The second is a virama, which removes the inherent vowel which occurs with consonants. The long underlines correspond to long vowels. Putting everything together, we obtain the phrase

kim japan mucyate ___ janma samsāra bandhanāt.

Googling (in Sanskrit or English) yields that this is part of a famous phrase in the Vishnu Sahasranāma (see e.g.Wikipedia), a Sanskrit hymn which is also one of the most popular stotras in modern Hinduism, and that the missing word is jantuh. In Sanskrit, this is जन्तुः.

Several sources have the missing word instead as जन्तुर (jantur). The extraction image eliminates this ambiguity. We thank Siddharta Jayanti for proposing the Shiva Sutras as an ordering of Sanskrit sounds and the use of this sloka from the Visnu Sahasranāma as a cluephrase for the final answer.

Authors' Notes

Nishant: I mainly worked on the Chaturanga puzzles. It was fun thinking of puzzles that look ridiculous or even illegal under regular chess rules, but are valid with the vastly weaker queen and bishop pieces in Chaturanga. It was a bit tricky to make sure that our positions were legal, since the queens are restricted to one color of square (if we imagine the boards as being checkered), and the black queen starts on e8 rather than d8. Even worse, with the bishop’s (2, 2) move, any particular bishop can only reach 8 out of the 64 squares of the board! Since white’s and black’s bishops start on ranks of different parity, they can never capture each other, and a particular color of bishop can only be on a quarter of the board’s squares.

I originally started with mate in one puzzles, with jumbled sequences of Aksharapalli numbers labeling the ranks and files. We eventually decided it would be cleaner for the puzzles to be series mates in two, so that we could instead extract two numbers directly from the board squares.

Overall, it was really cool to learn about all these different number systems using Sanskrit letters, especially how different number systems had varying applications in different fields of knowledge and culture. In fact, I think the Ascent round was my favorite round in this hunt, because testsolving and factchecking these puzzles allowed me to learn a lot about different languages and cultures around the world.

Abhijit: My parents are Carnatic musicians, and I was named after the “missing” nakshatra Abhijit, so I am very pleased that I was able to sneak these aspects of my life into a puzzle :) My sister drew the art for the Ramayana minipuzzle on very short notice, and drew several other images that didn’t end up making it into the final version of the puzzle. Here are all of them - see if you can figure out what parts of the Ramayana they correspond to!